ACTIVITY 6.2 CHOICES 713 OBJECTIVES ACTIVITY 6.2 Choices 1. Apply the multiplication principle of counting. 2. Determine the sample space for a probability distribution. 3. Display a sample space with a tree diagram. 4. Determine complementary probabilities. 5. Use Venn diagram's to illustrate relationships between events. Suppose you are looking for a new car and have narrowed your decision down to a Mustang, but can't decide on the exact color, transmission, engine, or options package. There are three sizes of engine (3.0 liters, 3.8 liters, and 4.6 liters), two transmissions (standard and automatic), five colors you like (black, silver, red, yellow, and green), and three option packages (GL, Sport, and XL). With all these possible choices, you want to know how many different Mustangs there are from which you must choose. 1. From the choices given, how many different Mustangs are possible? (Solve this any way you can. A diagram or list may help.) In solving the Mustang problem, you might have recognized a counting shortcut. If the outcomes you wish to count consist of a sequence of choices, it is possible to apply the multiplication principle of counting. For example, suppose you have three sweaters: one cotton, one wool, and one alpaca. You also have four hats, colored red, green, black, and purple. If you want to wear one sweater and one hat, how many different combinations are possible? Option 1: You could simply list all the possibilities: cotton red cotton + green cotton + black cotton + purple wool + red wool + green wool + black wool + purple alpaca + red alpaca + green alpaca + black alpaca + purple You could shorten the process with a tree diagram (and abbreviations), where each level of branching represents the next choice. Sweater Hat Outcome -r cr eg. cotton -^^7 ü r -b cb -P cp -r wr o Wg wool ^^m,, -b wb P wp r ar alpaca «é^ctu *! - b ab -P a P The total number of end points represents all the possibilities. Tracing each branch displays all the possible outcomes. Option 2: You could use a table approach if there are only two levels of choices. COTTON WOOL ALPACA cr wr ar GREEN BLACK PURPLE eg cb cp wg wb wp ag ab ap
714 CHAPTER 6 PROBABILITY MODELS Option 3: You could multiply the 3 choices for a sweater by the 4 choices for a hat to get a total of 12 combinations. This is called the multiplication principle of counting. The Multiplication Principle of Counting If some choice can be made in M ways and a subsequent choice can be made in N ways, then there are M times N ways these choices can be made in succession. 2. Apply the multiplication principle to the Mustang problem to verify your answer in Problem 1. 3. a. You want to create an ID code for all your customers based on three characters. The first character must be a letter of the alphabet, and the second and third must each be a digit between 1 and 9, inclusive. How many such codes are there? (Use the multiplication principle.) b. Would a list or tree diagram be practical in this case? 4. a. Suppose you flip a penny and a dime. Determine the number of outcomes. b. Use a tree diagram to display all possible outcomes. c. Assume that each single outcome is equally likely. Use the tree diagram to help calculate probabilities. What is the probability of getting two heads? d. What is the probability of getting one head and one tail, on either coin? e. If the variable x represents the number of heads, complete the table for this J probability distribution. 0 1 2
ACTIVITY 6.2 CHOICES 715 f. What must be true about the sum of the probabilities for this (and any other) probability distribution? The collection of all the possible single outcomes displayed by the tree diagram is the sample space for the probability distribution. 5. Two fair dice are rolled. a. Determine the number of possible outcomes. b. Display the sample space of all possible outcomes. You may use a tree diagram or a table. It may help to think of the dice as being different colors. c. What is the probability of getting a total of two dots on the dice (snake eyes)? That is, what is the probability of rolling a 1 on each die? d. What is the probability of rolling a total of 6 on the two dice? e. If the variable x represents the total number of spots on the two dice, refer to your sample space to complete the table for this probability distribution. x 2 3 4 5 7 f 10 11 12 POO f. What must be the sum of all probabilities? g. Are you as likely to roll a sum of 2 as you are of rolling a sum of 7? Explain. 6. To win the jackpot in a large state lottery, the winner must pick correctly all six numbers from among 1 through 54. The theoretical probability of doing this is x&jjâ «.0000000387. Stated another way, a single pick has a 1 in 25,827,165
716 CHAPTER 6 PROBABILITY MODELS chance to win the jackpot (and even then it might be shared with other winners). The theory of probability originated out of a desire to understand games of chance. Show how the 25,827,165 is determined. 7. What is the probability of getting ten heads on ten flips of a coin? 8. Two tennis balls are randomly selected from a bag that contains one Penn, one Wilson, and one Dunlop tennis ball. a. If the first tennis ball picked is replaced before the second tennis ball is selected, determine the number of outcomes in the sample space. b. Construct a tree diagram and list the outcomes in the sample space. c. Determine the probability that a Penn followed by a Wilson tennis ball is the outcome. d. Suppose the first tennis ball picked is not replaced before the second tennis ball is selected. Determine the number of outcomes in the sample space.
ACTIVITY 6.2 CHOICES 717 e. Construct a tree diagram and list the outcomes in the sample space. f. Determine the probability that a Penn followed by a Wilson tennis ball is the outcome. Complementary Events In some situations, you are interested in determining the probability that an event A does not happen. The event "not A", denoted by A, is called the complement of A. For example, The complement of success is failure. The complement of selected voter is Democrat is the selected voter is not a Democrat. The complement of a 6 is rolled is a 6 is not rolled. Since the sum of the probabilities for all outcomes of an experiment is 1, it follows that Rewriting, you have P(A) + P(A~) = 1. P(A)-= 1 - P(A) orp(a~) = 1 - P(A). 9. a. A die is rolled. What is the probability that the number 6 does not show? b. A card is randomly selected from a standard deck of 52 cards. What is the probability that the card is not a king?
718 CHAPTER 6 PROBABILITY MODELS Complementary probabilities are very useful when trying to determine the probability of "at least one." The statement that "an event happens at least once" is equivalent to "the event happens one or more times". Therefore, /event happens\ \at least once / /event happens \one or times (1) The complement of "an event happens one or more times" is "the event does not happen". Therefore, /event happens \, P\ J + /'(event does not happen) = 1 \one or more times (2) or /event happens \ P[. = 1 - /'(event does not happen). \one or more times/ (3) Substituting the results of equation (3) into equation (1), you have the formula /eventhappens\. P\ 1 = 1- F(event does not happen). Vat least once / 10. a. A fair dice is rolled. What is the probability that the number is at least two? b. Two fair die are rolled. What is the probability that the sum is at least 3? Venn Diagrams Complementary events can be illustrated using Venn diagrams. These diagrams were invented by English mathematician John Venn (1834-1923) and first appeared in a book on symbolic logic in 1881. In a Venn diagram, a rectangle generally represents the sample space. The items (all possible outcomes of a particular experiment) inside the rectangle can be separated into subdivisions which are generally represented by circles. The items in a circle represent the outcomes in the sample space that make a certain event true. For example, suppose a die is rolled. The rectangle would contain all the possible outcomes of whole numbers 1 through 6. If A represents the event that the number rolled is greater than 4, then the Venn diagram at the top of the next page represents the relationship between the sample space and the event A.
ACTIVITY 6.2 CHOICES 719 11. a. Determine the complement of event A in the roll-a-die experiment. b. Describe what portion of the Venn diagram above represents the complement of A. SUMMARY Activity 6.2 1. The multiplication principle of counting says that if some choice can be made in M ways and a subsequent choice can be made in N ways, then there are M times N ways these choices can be made in succession. 2. A tree diagram displays all possible outcomes for a sequence of choices, one outcome for each branch of the tree. 3. The sample space of a probability distribution is the collection of all possible outcomes. 4. The sum of the probability that an event A will occur and the probability that the event will not occur is 1. The event "not A," denoted by A, is called the complement of A. Stated symbolically, P(A) + P(A) = 1 or P(A) = 1 - P(A~). EXERCISES Activity 6.2 1. Phone numbers consist of a three-digit area code followed by seven digits. If the area eode must have a 0 or 1 for the second digit, and neither the area code nor the sevendigit number can start with 0 or 1, how many different phone numbers are possible? 2. You have four sweaters,fivepairs of pants, and three pairs of shoes. How many different combinations can you make, wearing one of each? 3. If you flip a coin ten times, how many different sequences of heads and tails are possible? 4. If you roll a die three times, how many different sequences are possible?